MEN OF MATHEMATICS did not spend all his time at Berlin but moved about. Part of his course was pursued at the University of Bonn, where his old teacher and friend Kumrner had taken the chair of mathe- matics. During Kronecker1 s residence at Bonn the University authorities were in the rnidst of a futile war to suppress the student societies whose chief object was the fostering of drink- ing, duelling, and brawling in general. With his customary astuteness, Kronecker allied himself secretly with the students and thereby made many friends who were later to prove useful. Kronecker's dissertation, accepted by Berlin for his Ph.D. in 1845, was inspired by Hummer's work in the theory of numbers and dealt with the units in certain algebraic number fields. Although the problem is one of extreme difficulty when it comes to actually exhibiting the units3 its nature can be understood from the following rough description of the general problem of units (for any algebraic number field, not merely for the special fields which interested Kummer and Kronecker). This sketch may also serve to make more intelligible some of the allusions in the present and subsequent chapters to the work of Kummer, Kronecker, and Dedekind in the higher arithmetic. The matter is quite simple but requires several preliminary definitions. The common whole numbers 1,2,3, ... are called the (posi- tive) rational integers. If m is any rational integer, it is the root of an algebraic equation of the first degree, whose coefficients are rational integers, namely x — m = 0. This, among other properties of the rational integers, suggested the generalization of the concept of integers to the 'numbers' defined as roots of algebraic equations. Thus if r is a root of the equation xn + aX1"1 + ... + an^x + an « 0, where the a's are rational integers (positive or negative), and if further r satisfies no equation of degree less than n, all of whose coefficients are rational integers and whose leading co- efficient is 1 (as it is in the above equation, namely the coeffi- cient of the highest power, xn9 of x in the equation is 1), then r is called an algebraic integer of degree ra. For example, 1 -f- V — 5 is an algebraic integer of degree 2, because it is a root of a? — 2az -f 6 = 0, and is not a root of any equation of degree 518